System described by differential equation

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System described by differential equation—
(i) d y(t) + 2y(t) = x(t)
dt

(ii) d y(t) + y(t) + 4 = x(t)
dt

1. (i) is linear but (ii) is non-linear
2. (i) is non-linear but (ii) is linear
3. both (i) and (ii) are linear
4. both (i) and (ii) are non-linear

Correct Option: A

(i) Let the response of the system to x₁(t) be y₁(t) and the response of the system to x₂(t) be y₂(t).
Thus, for the input x₁(t) the describing equation is

	dy₁(t)	+ 2y₁(t) = x₁(t) = F[y₁(t)]
	dt

and for the input

	dy₂(t)	+ 2y₂(t) = x₂(t) = F[y₂(t)]
	dt

Multiplying these equations by a₁ and a₂ respectively, and adding yields,

a₁	dy₁(t)	+ a₂	dy₂(t)	+ 2a₁y₁(t) + 2a₂y₂(t) = a₁x₁(t) + a₂x₂(t)
	dt		dt

	d	[a₁y₁(t) + a₂y₂(t)] + 2 [a₁y₁(t) + a₂y₂(t)] = a₁x₁(t) + a₂x₂(t)
	dt

i.e., F[y₁(t)] + F[y₂(t)] = F[y₁(t) + y₂(t)]
Hence, the system is linear. (ii) Similarly, for equation

	d	y(t) + 4 = x(t)
	dt

	dy₁(t)	+ y₁(t) + 4 = x₁(t) = F[y₁(t)]
	dt

and for input x₂(t).
Multiplying these equations by a₁ and a₂ respectively, and adding yields,

	dy₂(t)	+ y₂(t) + 4 = x₂(t) = F[y₂(t)]
	dt

i.e.,	d	+ [a₁y₁(t) + a₂y₂(t)] + 2 [a₁y₁(t) + a₂y₂(t)] + 4(a₁ + a₂) ≠ a₁.x₁(t) + a₂.x₂(t)
	dt

i.e., F[y₁(t)] + F[y₂(t)] ≠ F[y₁(t) + y₂(t)]
Hence, the system is non-linear.
Therefore, alternative (A) is the correct choice.